A new section of The Tau Manifesto: Getting to the bottom of pi(tauday.com)
tauday.com
A new section of The Tau Manifesto: Getting to the bottom of pi
http://tauday.com/tau-manifesto?new
12 comments
"pi is a lie" and "pi is wrong" are incredibly disingenuous slogans.
For the former, you just need to know your meme:
http://www.urbandictionary.com/define.php?term=the+cake+is+a...
For the latter, you need to know that it comes from Bob Palais' original article.
For both, you might need to lighten up.
For the former, you just need to know your meme:
http://www.urbandictionary.com/define.php?term=the+cake+is+a...
For the latter, you need to know that it comes from Bob Palais' original article.
For both, you might need to lighten up.
I think you need to dial up your humor level a bit when reading this.
The last section nails it in a nutshell:
...imagine we lived in a world where we used the letter h to represent “one half” and had no separate notation for 2h. We would then observe that h is ubiquitous in mathematics. In fact, 2h is the multiplicative identity, so how can one doubt the importance of h? But this is crazy: 2h is the fundamental number, not h. Let us therefore introduce a separate symbol for 2h; call it 1. We then see that h=1/2, and there is no longer any reason to use h at all.
Color me converted as well.
...imagine we lived in a world where we used the letter h to represent “one half” and had no separate notation for 2h. We would then observe that h is ubiquitous in mathematics. In fact, 2h is the multiplicative identity, so how can one doubt the importance of h? But this is crazy: 2h is the fundamental number, not h. Let us therefore introduce a separate symbol for 2h; call it 1. We then see that h=1/2, and there is no longer any reason to use h at all.
Color me converted as well.
It seems like these guys are fighting an uphill battle, not that I'd want to discourage it. It's simply much easier to visualize the path traced when you relate the radius to the circumference of a circle rather than its diameter, since there's only one point to focus on at a time as you rotate around the axis. Pi is insidious as it's only much later that this foundation begins to appear problematic.
[My analogy was not very good. The circumference is the only dimension of a circle stated in terms of diameter, not radius, and so substituting tau for pi is simply relating the radius of a circle to its circumference instead of its diameter. Circumference = tau * radius. This has the benefit of bringing the circumference formula in line with all the other circle equations.
When we look at the area of a circle, tau is not very intuitive, however. The formula for the area of a circle uses the square of the radius, so there's not an intuitive way to factor two into the equation without just making extra mental work for yourself.
For volume, the effect is somewhat neutral. Volume = 4/3 pi * radius^3 = 2/3 tau * r^3. 2/3 vs. 4/3 doesn't seem to add much insight into the equation.
Circumference = tau * r
Area = 1/2 tau * r^2
Volume = 2/3 tau * r^3
4th dimension circle = 3/4 tau * r^4? : P
Maybe there's something to this...]
[My analogy was not very good. The circumference is the only dimension of a circle stated in terms of diameter, not radius, and so substituting tau for pi is simply relating the radius of a circle to its circumference instead of its diameter. Circumference = tau * radius. This has the benefit of bringing the circumference formula in line with all the other circle equations.
When we look at the area of a circle, tau is not very intuitive, however. The formula for the area of a circle uses the square of the radius, so there's not an intuitive way to factor two into the equation without just making extra mental work for yourself.
For volume, the effect is somewhat neutral. Volume = 4/3 pi * radius^3 = 2/3 tau * r^3. 2/3 vs. 4/3 doesn't seem to add much insight into the equation.
Circumference = tau * r
Area = 1/2 tau * r^2
Volume = 2/3 tau * r^3
4th dimension circle = 3/4 tau * r^4? : P
Maybe there's something to this...]
No, the tau manifesto agrees: radius is the more fundamental concept. That's why tau is better! Pi is an expression relating the diameter to the circumference, which as you point out is a little counterintuitive.
The problem with tau for introductory math is area = pi * r^2 = (tau/2) * r^2. Given the choice between having equations with 2x or other with 1/2 y most people feel more comfortable with 2x.
EX: e^(i * tau / 2) = -1 wait what?
EX: e^(i * tau / 2) = -1 wait what?
If you read the Tau Manifesto you'll see why tau * r^2 / 2 is actually AWESOME (he actually refers to it as pi's coup de grace)
Just think of it like an integral (that's what it is, after all!), raise the power and divide by the power. Suddenly the origin of this formula is no longer obscured.
Regarding Euler's identity: e^(i * tau) = 1
Just think of it like an integral (that's what it is, after all!), raise the power and divide by the power. Suddenly the origin of this formula is no longer obscured.
Regarding Euler's identity: e^(i * tau) = 1
Telling a 3rd grader to think of it like an integral seems to be putting the cart WAAAAAAAY before the horse.
Edit: Also that's is hardly a more fundamental equation because you could also say e^(1024 pi * i) = e^(4096 * tau * i) = 1.
Edit: Also that's is hardly a more fundamental equation because you could also say e^(1024 pi * i) = e^(4096 * tau * i) = 1.
Eh, 3rd graders can probably handle the area of a triangle, which can be viewed as a definite integral. (Somehow, middle school physics students seem to be able to handle computing the distance traveled by an object given starting velocity and constant acceleration--again, technically an integral.)
In fact, one can view a circle as being made up of a bunch of really thin triangles (this is how one usually derives the area of a circle), and this again is something that third-graders can appreciate--cut a paper circle into slices, and see if you can get the slices to look more and more like triangles as you cut them smaller. (I vaguely remember doing something like this in elementary school.)
In fact, one can view a circle as being made up of a bunch of really thin triangles (this is how one usually derives the area of a circle), and this again is something that third-graders can appreciate--cut a paper circle into slices, and see if you can get the slices to look more and more like triangles as you cut them smaller. (I vaguely remember doing something like this in elementary school.)
It won't amaze third graders, but it would make a lot more sense the first time you learn integral calculus
Meh, so did 2 * pi * r > 2 * pi * (r ^ 2) / 2 = pi * r ^ 2.
The great thing about math is it's it works out either way. So, it really comes down to which is easier to deal with and students seem to have an easier time remembering and understanding 2pir and pi*r^2, but in the end it's tau day so that's what people want to argue about.
The great thing about math is it's it works out either way. So, it really comes down to which is easier to deal with and students seem to have an easier time remembering and understanding 2pir and pi*r^2, but in the end it's tau day so that's what people want to argue about.
"students seem to have an easier time remembering and understanding 2 pi r and pi r^2"
Are you speaking from empirical experience as a math teacher?
What makes 2 pi r easier to understand than tau r?
Are you speaking from empirical experience as a math teacher?
What makes 2 pi r easier to understand than tau r?
Also perhaps, make integral calculus make more sense?
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I wonder if the fourth dimension in this case might be the maximum internal surface area or something of the sort.
A fourth dimension perfect circle looks like a sphere at first glance, but has the maximum possible internal surface area upon closer inspection.
A fourth dimension perfect circle looks like a sphere at first glance, but has the maximum possible internal surface area upon closer inspection.
I live with 60 minutes per hour, 24 hours per day, 7 days per week, 360° in a full circle, a speed of light of 299,792,458 m/s, to name only a few cases of historical 'accidents'.
2π is arguably more fundamental than π, but given the things above, I don't care much.
If you want to tackle something worthwhile, update theoretical physics lectures to 'modern' notation (where modern means 1960s).
2π is arguably more fundamental than π, but given the things above, I don't care much.
If you want to tackle something worthwhile, update theoretical physics lectures to 'modern' notation (where modern means 1960s).
360° in a full circle
That's one of the things tau can help solve. I suspect the main reason that people still use degrees so much is because using pi undermines the beauty and intuitiveness of radians.
That's one of the things tau can help solve. I suspect the main reason that people still use degrees so much is because using pi undermines the beauty and intuitiveness of radians.
The main reason people still use degrees is because the majority of the population only know degrees, can't be bothered to change, and the majority of the population will continue to only know degrees, unless radians start getting taught at a much younger age.
People can't be bothered to change, except radians are much more mathematically easy to use, so people who need to do advanced maths will use it.
However, laziness is also the reason why people don't change to tau. tau is just a multiple of pi. Confronted with such a small change, people will just think that they can't be bothered to change, especially since pi is already widely supported and implemented in any of the tools that we use, and tau is not.
There's no way that changing to tau will make more people use radians.
People can't be bothered to change, except radians are much more mathematically easy to use, so people who need to do advanced maths will use it.
However, laziness is also the reason why people don't change to tau. tau is just a multiple of pi. Confronted with such a small change, people will just think that they can't be bothered to change, especially since pi is already widely supported and implemented in any of the tools that we use, and tau is not.
There's no way that changing to tau will make more people use radians.
No, the main reason why degrees are still in use is because most people have an easier time with 51° instead of 1/7 τ.
51° is not 1/7 τ. 1/7 τ is (51 3/7)°, or 51.428571...°. If you round that to 51°, then presumably you're doing something where inexact numbers are fine, in which case you might then use "0.89 (radians)". My intuition is better for degrees here, but I think that would change if I spent ten minutes doing calculations involving real radian measures.
51° is the latitude of the city I'm living in, and I chose 1/7 τ because it comes pretty close.
I could have used 0.14 τ to make my point as well: If you want to avoid fractions in common cases, you'll need to multiply with an arbitrary number like, say 100 (which has obvious benefits in a base-10 system), but arguments could be made for something like 400 (in which case we end up with gon) or, say 360, which has the benefit that it's the established standard (even though it's not the most obvious choice).
Actual radians (ie not expressed as fractions of 2π) are pretty user-unfriendly in a base-10 system.
I could have used 0.14 τ to make my point as well: If you want to avoid fractions in common cases, you'll need to multiply with an arbitrary number like, say 100 (which has obvious benefits in a base-10 system), but arguments could be made for something like 400 (in which case we end up with gon) or, say 360, which has the benefit that it's the established standard (even though it's not the most obvious choice).
Actual radians (ie not expressed as fractions of 2π) are pretty user-unfriendly in a base-10 system.
> Actual radians (ie not expressed as fractions of 2π)
I think this is the source of your problem: you view an expression like "τ/8" or "π/4" as less an "actual" radian measure than the inexact ".785". That's a function of how a lot of us were taught math, I think, and I'm quite sure that it's only made worse by the fact that the use of π camouflages the fact that this is related to fractions of a circle. But your premise is faulty. Of course radians expressed in decimal are user-unfriendly, but radians expressed in decimal are no less "actual" than radians expressed as fractions of τ or π.
I think this is the source of your problem: you view an expression like "τ/8" or "π/4" as less an "actual" radian measure than the inexact ".785". That's a function of how a lot of us were taught math, I think, and I'm quite sure that it's only made worse by the fact that the use of π camouflages the fact that this is related to fractions of a circle. But your premise is faulty. Of course radians expressed in decimal are user-unfriendly, but radians expressed in decimal are no less "actual" than radians expressed as fractions of τ or π.
A physical quantity is given by numerical value and unit of measurement. Giving angles as fractions of τ changes the unit of measurement from radians to turn.
There are applications for which neither radians nor turn are a good fit. Similarly, oftentimes the 'messy' SI units are a far better fit than any 'more fundamental' system of natural units.
There are applications for which neither radians nor turn are a good fit. Similarly, oftentimes the 'messy' SI units are a far better fit than any 'more fundamental' system of natural units.
I, for one, speak in terms of degrees as a sort of compatibility mode--when talking to people who aren't familiar with tau and when I don't feel like pressing the point with them. (I have almost completely avoided mentioning "pi" except in meta-discussions since I read the Tau Manifesto two years ago.)
"For example, consider integrals over all space in polar coordinates:
\[ \int_0^{2\pi}\int_0^\infty f(r, \theta) r dr d\theta \]
The upper limit of the $\theta$ integration is always 2$\pi$."
This statement is false. If $r = r(\theta)$, then it can be the case (e.g., for a "looped" limacon $r = \cos(\theta)$) that $\theta$ will not range from $[0,2\pi]$. The upper limit is not necessarily "always" $2\pi$.
\[ \int_0^{2\pi}\int_0^\infty f(r, \theta) r dr d\theta \]
The upper limit of the $\theta$ integration is always 2$\pi$."
This statement is false. If $r = r(\theta)$, then it can be the case (e.g., for a "looped" limacon $r = \cos(\theta)$) that $\theta$ will not range from $[0,2\pi]$. The upper limit is not necessarily "always" $2\pi$.
This should probably link to the actual section of the page that was added: http://tauday.com/tau-manifesto?new#sec:getting_to_the_botto...
The reason I didn't link to the exact section is because the math needs some time to render. If you go right to the section, on many browsers you won't end up where you wanted to be. If instead people hit the top of the page, the math renders in the background while they're looking for the new section link.
Huh. Section 5.1 (one of the new ones) is presented as this new epiphany about hyperspheres and tau, but I made a post back in 2010 about this ( http://www.blahedo.org/blog/archives/001083.html ), emailed Hartl at the time, and he said he'd already thought of that. Still, cool to see that my observation was worth (eventually) including in the manifesto itself. (He did elaborate on the idea considerably, of course.) ;)
When I said I'd already thought of it, I was referring to the general observation that n-sphere volumes support tau, not pi. You can find my original comments, dating from July 2010, at http://forums.xkcd.com/viewtopic.php?f=17&t=61958.
The "epiphany" refers to the specific realization that there are three families of constants, and in particular that pi is not a member of the family of volume constants but rather is part of a useless family of its own.
In any case, you certainly deserve mention, so I've added you to the acknowledgments. Thanks!
The "epiphany" refers to the specific realization that there are three families of constants, and in particular that pi is not a member of the family of volume constants but rather is part of a useless family of its own.
In any case, you certainly deserve mention, so I've added you to the acknowledgments. Thanks!
Nifty! I wasn't trolling for recognition so much as being puzzled why this was billed as a new epiphany---that makes sense though. I love the idea of recasting the presentation from "list of complex formulae for area and volume" to "sequence of constants expressing ratios".
I know you weren't trolling, but I'm a strong believer in giving credit where credit is due. Thanks again for your comments!
I know that as a kid I was all too sensitive to things in math "making sense" in a deep way. Any semi-arbitrary conventions that made my understanding less elegant, or which obscured the beauty of a concept, were fiercely protested.
So, up with tau!
So, up with tau!
[deleted]
It's just semantics. Pi and tau are the same thing. You don't see physicists going around saying h is wrong and h-bar is what they need to use.
You don't see physicists going around saying h is wrong and h-bar is what they need to use.
Sure they do—or at least, they did. Physicists realized that h is "wrong", i.e., confusing and unnatural, because it's off by a factor of, um, 2 pi. They introduced h-bar precisely to rectify the problem. And the substitution worked: if you open up a standard book on quantum mechanics, the ratio of uses of h to h-bar is just about epsilon. (You'll also see lots of (2 pi)s.)
Sure they do—or at least, they did. Physicists realized that h is "wrong", i.e., confusing and unnatural, because it's off by a factor of, um, 2 pi. They introduced h-bar precisely to rectify the problem. And the substitution worked: if you open up a standard book on quantum mechanics, the ratio of uses of h to h-bar is just about epsilon. (You'll also see lots of (2 pi)s.)
"It's just semantics" is a horrible, useless phrase.
The difference between "robot" and "postlapsarian" is "just semantics", after all.
The difference between "robot" and "postlapsarian" is "just semantics", after all.
Happy Tau Day
Am in and converted!
pi is not a lie. It's not part of some conspiracy theory among math elites to mislead the masses. It's not about math bureaucrats refusing to admit they were wrong. Everything they told you about pi is true (or it isn't, but in that case it's not true of tau either). pi and tau are really just two different notations for the same concept. Only perhaps one is more elegant and convenient than the other, making it more fundamental in some mystical way (and perhaps preferable if you ever need to convince visiting aliens that you are intelligent).